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Aims. We investigated plasma turbulence in the context of solar wind. We concentrated on properties of ideal second-order magneto-hydrodynamic (MHD) and Hall MHD invariants. Methods. We studied the results of a two-dimensional hybrid simulation of decaying plasma turbulence with an initial large cross helicity and a negligible magnetic helicity. We investigated the evolution of the combined energy and the cross, kinetic, mixed, and magnetic helicities. For the combined (kinetic plus magnetic) energy and the cross, kinetic, and mixed helicities, we analysed the corresponding Kármán-Howarth-Monin (KHM) equation in the hybrid (kinetic proton and fluid electron) approximation. Results. The KHM analysis shows that the combined energy decays at large scales. At intermediate scales, this energy cascades (from large to small scales) via the MHD non-linearity and this cascade partly continues via Hall coupling to sub-ion scales. The cascading combined energy is transferred (dissipated) to the internal energy at small scales via the resistive dissipation and the pressure-strain effect. The Hall term couples the cross helicity with the kinetic one, suggesting that the coupled invariant, referred to here as the mixed helicity, is a relevant turbulence quantity. However, when analysed using the KHM equations, the kinetic and mixed helicities exhibit very dissimilar behaviours to that of the combined energy. On the other hand, the cross helicity, in analogy to the energy, decays at large scales, cascades from large to small scales via the MHD+Hall non-linearity, and is dissipated at small scales via the resistive dissipation and the cross-helicity equivalent of the pressure-strain effect. In contrast to the combined energy, the Hall term is important for the cross helicity over a wide range of scales (even well above ion scales). In contrast, the magnetic helicity is scantily generated through the resistive term and does not exhibit any cascade.
Hellinger et al. (Thu,) studied this question.